# Ngô Quốc Anh

## March 5, 2014

### Uniformly upper Bound for Positive Smooth Solutions To The Lichnerowicz Equation In R^N

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 16:16

In this note, we are interested in the following Lichnerowicz type equation in $\mathbb R^n$

$\displaystyle -\Delta u =-u^q+\frac{1}{u^{q+2}}, \quad q>0.$

As we have already seen from the previous note that solutions for the above equation are always bounded from below for certain $q$.

Theorem (Brezis). Any solution of the Lichnerowicz equation with $q>0$ satisfies $u\geqslant 1$ in $\mathbb R^n$.

Remarkably, we are able to prove that solutions for the Lichnerowicz type equation are also bounded from above if we require $q>1$ instead of $q>0$. This result is basically due to L. Ma and X. Xu, see this paper.

Theorem (Ma-Xu). Any solution of the Lichnerowicz type equation with $q>1$ is uniformly bounded from above in $\mathbb R^n$.

Combining the two theorem above, we conclude that any solution of the Lichnerowicz equation, i.e. $q=(n+2)/(n-2)$, is uniformly bounded in $\mathbb R^n$.

The idea of the proof for Ma-Xu’s theorem is as follows: Denote

$\displaystyle f(u)=u^{-q-2} - u^q.$

Fix $x_0 \in \mathbb R^n$ but arbitrary, we then look for a positive radial super-solution $v(x)=v(|x|)>0$ of the Lichnerowicz type equation on the ball $B_R(x_0)$ with positive infinity boundary condition for some $R$ to be specified. This is equivalent to finding $v$ in such a way that

$\displaystyle\begin{array}{rcl}\displaystyle-\Delta v &\geqslant& f(v), \qquad\text{ in } B_R(x_0),\\ v&\equiv& +\infty, \qquad\text{ on }\partial B_R(x_0).\end{array}$